A154246 a(n) = ( (5 + sqrt(7))^n - (5 - sqrt(7))^n )/(2*sqrt(7)).

1, 10, 82, 640, 4924, 37720, 288568, 2206720, 16872976, 129008800, 986374432, 7541585920, 57661119424, 440862647680, 3370726327168, 25771735613440, 197044282245376, 1506551581411840, 11518718733701632, 88069258871603200

Offset: 1

Author

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

Keywords

• nonn
• easy

Comments

Second binomial transform of A086901 without initial term 1.

Lim_{n -> infinity} a(n)/a(n-1) = 5 + sqrt(7) = 7.6457513110....

Links

• G. C. Greubel, Table of n, a(n) for n = 1..750
• Index entries for linear recurrences with constant coefficients, signature (10,-18).

Crossrefs

Cf. A010465 (decimal expansion of square root of 7), A086901.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[((5+r)^n-(5-r)^n)/(2*r): n in [1..25]]; [Integers()!S[j]: j in [1..#S]]; // Klaus Brockhaus, Jan 07 2009

Magma

I:=[1,10]; [n le 2 select I[n] else 10*Self(n-1)-18*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Sep 08 2016

Mathematica

Table[Simplify[((5+Sqrt[7])^n -(5-Sqrt[7])^n)/(2*Sqrt[7])], {n,1,25}] (* or *) LinearRecurrence[{10, -18}, {1, 10}, 25] (* G. C. Greubel, Sep 07 2016 *)

PARI

my(x='x+O('x^25)); Vec(x/(1-10*x+18*x^2)) \\ G. C. Greubel, May 31 2019

Sage

[lucas_number1(n,10,18) for n in range(1, 25)] # Zerinvary Lajos, Apr 26 2009

Formula

From Philippe Deléham, Jan 06 2009: (Start)

a(n) = 10*a(n-1) - 18*a(n-2) for n > 1, with a(0)=0, a(1)=1.

G.f.: x/(1 - 10*x + 18*x^2). (End)

E.g.f.: (1/sqrt(7))*exp(5*x)*sinh(sqrt(7)*x). - G. C. Greubel, Sep 07 2016

Extensions

Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009

Edited by Klaus Brockhaus, Oct 06 2009